// Copyright (c) 2021, gottingen group.
// All rights reserved.
// Created by liyinbin lijippy@163.com

#include "testing/distribution_test_util.h"
#include "abel/log/logging.h"
#include "abel/base/profile.h"
#include "abel/strings/str_cat.h"
#include "abel/strings/format.h"
#include <cassert>
#include <cmath>
#include <string>
#include <vector>

namespace abel {

namespace random_internal {
namespace {

#if defined(__EMSCRIPTEN__)
// Workaround __EMSCRIPTEN__ error: llvm_fma_f64 not found.
ABEL_FORCE_INLINE double fma(double x, double y, double z) { return (x * y) + z; }
#endif

}  // namespace

DistributionMoments ComputeDistributionMoments(
        abel::span<const double> data_points) {
    DistributionMoments result;

    // Compute m1
    for (double x : data_points) {
        result.n++;
        result.mean += x;
    }
    result.mean /= static_cast<double>(result.n);

    // Compute m2, m3, m4
    for (double x : data_points) {
        double v = x - result.mean;
        result.variance += v * v;
        result.skewness += v * v * v;
        result.kurtosis += v * v * v * v;
    }
    result.variance /= static_cast<double>(result.n - 1);

    result.skewness /= static_cast<double>(result.n);
    result.skewness /= std::pow(result.variance, 1.5);

    result.kurtosis /= static_cast<double>(result.n);
    result.kurtosis /= std::pow(result.variance, 2.0);
    return result;

    // When validating the min/max count, the following confidence intervals may
    // be of use:
    // 3.291 * stddev = 99.9% CI
    // 2.576 * stddev = 99% CI
    // 1.96 * stddev  = 95% CI
    // 1.65 * stddev  = 90% CI
}

std::ostream &operator<<(std::ostream &os, const DistributionMoments &moments) {
    return os << abel::sprintf("mean=%f, stddev=%f, skewness=%f, kurtosis=%f",
                               moments.mean, std::sqrt(moments.variance),
                               moments.skewness, moments.kurtosis);
}

double InverseNormalSurvival(double x) {
    // inv_sf(u) = -sqrt(2) * erfinv(2u-1)
    static constexpr double kSqrt2 = 1.4142135623730950488;
    return -kSqrt2 * abel::random_internal::erfinv(2 * x - 1.0);
}

bool Near(std::string_view msg, double actual, double expected, double bound) {
    assert(bound > 0.0);
    double delta = fabs(expected - actual);
    if (delta < bound) {
        return true;
    }

    std::string formatted = abel::string_cat(
            msg, " actual=", actual, " expected=", expected, " err=", delta / bound);
    DLOG_INFO("{}", formatted.c_str());
    return false;
}

// TODO(abel-team): Replace with an "ABEL_HAVE_SPECIAL_MATH" and try
// to use std::beta().  As of this writing P0226R1 is not implemented
// in libc++: http://libcxx.llvm.org/cxx1z_status.html
double beta(double p, double q) {
    // Beta(x, y) = Gamma(x) * Gamma(y) / Gamma(x+y)
    double lbeta = std::lgamma(p) + std::lgamma(q) - std::lgamma(p + q);
    return std::exp(lbeta);
}

// Approximation to inverse of the Error Function in double precision.
// (http://people.maths.ox.ac.uk/gilesm/files/gems_erfinv.pdf)
double erfinv(double x) {
#if !defined(__EMSCRIPTEN__)
    using std::fma;
#endif

    double w = 0.0;
    double p = 0.0;
    w = -std::log((1.0 - x) * (1.0 + x));
    if (w < 6.250000) {
        w = w - 3.125000;
        p = -3.6444120640178196996e-21;
        p = fma(p, w, -1.685059138182016589e-19);
        p = fma(p, w, 1.2858480715256400167e-18);
        p = fma(p, w, 1.115787767802518096e-17);
        p = fma(p, w, -1.333171662854620906e-16);
        p = fma(p, w, 2.0972767875968561637e-17);
        p = fma(p, w, 6.6376381343583238325e-15);
        p = fma(p, w, -4.0545662729752068639e-14);
        p = fma(p, w, -8.1519341976054721522e-14);
        p = fma(p, w, 2.6335093153082322977e-12);
        p = fma(p, w, -1.2975133253453532498e-11);
        p = fma(p, w, -5.4154120542946279317e-11);
        p = fma(p, w, 1.051212273321532285e-09);
        p = fma(p, w, -4.1126339803469836976e-09);
        p = fma(p, w, -2.9070369957882005086e-08);
        p = fma(p, w, 4.2347877827932403518e-07);
        p = fma(p, w, -1.3654692000834678645e-06);
        p = fma(p, w, -1.3882523362786468719e-05);
        p = fma(p, w, 0.0001867342080340571352);
        p = fma(p, w, -0.00074070253416626697512);
        p = fma(p, w, -0.0060336708714301490533);
        p = fma(p, w, 0.24015818242558961693);
        p = fma(p, w, 1.6536545626831027356);
    } else if (w < 16.000000) {
        w = std::sqrt(w) - 3.250000;
        p = 2.2137376921775787049e-09;
        p = fma(p, w, 9.0756561938885390979e-08);
        p = fma(p, w, -2.7517406297064545428e-07);
        p = fma(p, w, 1.8239629214389227755e-08);
        p = fma(p, w, 1.5027403968909827627e-06);
        p = fma(p, w, -4.013867526981545969e-06);
        p = fma(p, w, 2.9234449089955446044e-06);
        p = fma(p, w, 1.2475304481671778723e-05);
        p = fma(p, w, -4.7318229009055733981e-05);
        p = fma(p, w, 6.8284851459573175448e-05);
        p = fma(p, w, 2.4031110387097893999e-05);
        p = fma(p, w, -0.0003550375203628474796);
        p = fma(p, w, 0.00095328937973738049703);
        p = fma(p, w, -0.0016882755560235047313);
        p = fma(p, w, 0.0024914420961078508066);
        p = fma(p, w, -0.0037512085075692412107);
        p = fma(p, w, 0.005370914553590063617);
        p = fma(p, w, 1.0052589676941592334);
        p = fma(p, w, 3.0838856104922207635);
    } else {
        w = std::sqrt(w) - 5.000000;
        p = -2.7109920616438573243e-11;
        p = fma(p, w, -2.5556418169965252055e-10);
        p = fma(p, w, 1.5076572693500548083e-09);
        p = fma(p, w, -3.7894654401267369937e-09);
        p = fma(p, w, 7.6157012080783393804e-09);
        p = fma(p, w, -1.4960026627149240478e-08);
        p = fma(p, w, 2.9147953450901080826e-08);
        p = fma(p, w, -6.7711997758452339498e-08);
        p = fma(p, w, 2.2900482228026654717e-07);
        p = fma(p, w, -9.9298272942317002539e-07);
        p = fma(p, w, 4.5260625972231537039e-06);
        p = fma(p, w, -1.9681778105531670567e-05);
        p = fma(p, w, 7.5995277030017761139e-05);
        p = fma(p, w, -0.00021503011930044477347);
        p = fma(p, w, -0.00013871931833623122026);
        p = fma(p, w, 1.0103004648645343977);
        p = fma(p, w, 4.8499064014085844221);
    }
    return p * x;
}

namespace {

// Direct implementation of AS63, BETAIN()
// https://www.jstor.org/stable/2346797?seq=3#page_scan_tab_contents.
//
// BETAIN(x, p, q, beta)
//  x:     the value of the upper limit x.
//  p:     the value of the parameter p.
//  q:     the value of the parameter q.
//  beta:  the value of ln B(p, q)
//
double BetaIncompleteImpl(const double x, const double p, const double q,
                          const double beta) {
    if (p < (p + q) * x) {
        // Incomplete beta function is symmetrical, so return the complement.
        return 1. - BetaIncompleteImpl(1.0 - x, q, p, beta);
    }

    double psq = p + q;
    const double kErr = 1e-14;
    const double xc = 1. - x;
    const double pre =
            std::exp(p * std::log(x) + (q - 1.) * std::log(xc) - beta) / p;

    double term = 1.;
    double ai = 1.;
    double result = 1.;
    int ns = static_cast<int>(q + xc * psq);

    // Use the soper reduction forumla.
    double rx = (ns == 0) ? x : x / xc;
    double temp = q - ai;
    for (;;) {
        term = term * temp * rx / (p + ai);
        result = result + term;
        temp = std::fabs(term);
        if (temp < kErr && temp < kErr * result) {
            return result * pre;
        }
        ai = ai + 1.;
        --ns;
        if (ns >= 0) {
            temp = q - ai;
            if (ns == 0) {
                rx = x;
            }
        } else {
            temp = psq;
            psq = psq + 1.;
        }
    }

    // NOTE: See also TOMS Alogrithm 708.
    // http://www.netlib.org/toms/index.html
    //
    // NOTE: The NWSC library also includes BRATIO / ISUBX (p87)
    // https://archive.org/details/DTIC_ADA261511/page/n75
}

// Direct implementation of AS109, XINBTA(p, q, beta, alpha)
// https://www.jstor.org/stable/2346798?read-now=1&seq=4#page_scan_tab_contents
// https://www.jstor.org/stable/2346887?seq=1#page_scan_tab_contents
//
// XINBTA(p, q, beta, alhpa)
//  p:     the value of the parameter p.
//  q:     the value of the parameter q.
//  beta:  the value of ln B(p, q)
//  alpha: the value of the lower tail area.
//
double BetaIncompleteInvImpl(const double p, const double q, const double beta,
                             const double alpha) {
    if (alpha < 0.5) {
        // Inverse Incomplete beta function is symmetrical, return the complement.
        return 1. - BetaIncompleteInvImpl(q, p, beta, 1. - alpha);
    }
    const double kErr = 1e-14;
    double value = kErr;

    // Compute the initial estimate.
    {
        double r = std::sqrt(-std::log(alpha * alpha));
        double y =
                r - fma(r, 0.27061, 2.30753) / fma(r, fma(r, 0.04481, 0.99229), 1.0);
        if (p > 1. && q > 1.) {
            r = (y * y - 3.) / 6.;
            double s = 1. / (p + p - 1.);
            double t = 1. / (q + q - 1.);
            double h = 2. / s + t;
            double w =
                    y * std::sqrt(h + r) / h - (t - s) * (r + 5. / 6. - t / (3. * h));
            value = p / (p + q * std::exp(w + w));
        } else {
            r = q + q;
            double t = 1.0 / (9. * q);
            double u = 1.0 - t + y * std::sqrt(t);
            t = r * (u * u * u);
            if (t <= 0) {
                value = 1.0 - std::exp((std::log((1.0 - alpha) * q) + beta) / q);
            } else {
                t = (4.0 * p + r - 2.0) / t;
                if (t <= 1) {
                    value = std::exp((std::log(alpha * p) + beta) / p);
                } else {
                    value = 1.0 - 2.0 / (t + 1.0);
                }
            }
        }
    }

    // Solve for x using a modified newton-raphson method using the function
    // BetaIncomplete.
    {
        value = std::max(value, kErr);
        value = std::min(value, 1.0 - kErr);

        const double r = 1.0 - p;
        const double t = 1.0 - q;
        double y;
        double yprev = 0;
        double sq = 1;
        double prev = 1;
        for (;;) {
            if (value < 0 || value > 1.0) {
                // Error case; value went infinite.
                return std::numeric_limits<double>::infinity();
            } else if (value == 0 || value == 1) {
                y = value;
            } else {
                y = BetaIncompleteImpl(value, p, q, beta);
                if (!std::isfinite(y)) {
                    return y;
                }
            }
            y = (y - alpha) *
                std::exp(beta + r * std::log(value) + t * std::log(1.0 - value));
            if (y * yprev <= 0) {
                prev = std::max(sq, std::numeric_limits<double>::min());
            }
            double g = 1.0;
            for (;;) {
                const double adj = g * y;
                const double adj_sq = adj * adj;
                if (adj_sq >= prev) {
                    g = g / 3.0;
                    continue;
                }
                const double tx = value - adj;
                if (tx < 0 || tx > 1) {
                    g = g / 3.0;
                    continue;
                }
                if (prev < kErr) {
                    return value;
                }
                if (y * y < kErr) {
                    return value;
                }
                if (tx == value) {
                    return value;
                }
                if (tx == 0 || tx == 1) {
                    g = g / 3.0;
                    continue;
                }
                value = tx;
                yprev = y;
                break;
            }
        }
    }

    // NOTES: See also: Asymptotic inversion of the incomplete beta function.
    // https://core.ac.uk/download/pdf/82140723.pdf
    //
    // NOTE: See the Boost library documentation as well:
    // https://www.boost.org/doc/libs/1_52_0/libs/math/doc/sf_and_dist/html/math_toolkit/special/sf_beta/ibeta_function.html
}

}  // namespace

double BetaIncomplete(const double x, const double p, const double q) {
    // Error cases.
    if (p < 0 || q < 0 || x < 0 || x > 1.0) {
        return std::numeric_limits<double>::infinity();
    }
    if (x == 0 || x == 1) {
        return x;
    }
    // ln(Beta(p, q))
    double beta = std::lgamma(p) + std::lgamma(q) - std::lgamma(p + q);
    return BetaIncompleteImpl(x, p, q, beta);
}

double BetaIncompleteInv(const double p, const double q, const double alpha) {
    // Error cases.
    if (p < 0 || q < 0 || alpha < 0 || alpha > 1.0) {
        return std::numeric_limits<double>::infinity();
    }
    if (alpha == 0 || alpha == 1) {
        return alpha;
    }
    // ln(Beta(p, q))
    double beta = std::lgamma(p) + std::lgamma(q) - std::lgamma(p + q);
    return BetaIncompleteInvImpl(p, q, beta, alpha);
}

// Given `num_trials` trials each with probability `p` of success, the
// probability of no failures is `p^k`. To ensure the probability of a failure
// is no more than `p_fail`, it must be that `p^k == 1 - p_fail`. This function
// computes `p` from that equation.
double RequiredSuccessProbability(const double p_fail, const int num_trials) {
    double p = std::exp(std::log(1.0 - p_fail) / static_cast<double>(num_trials));
    ABEL_ASSERT(p > 0);
    return p;
}

double ZScore(double expected_mean, const DistributionMoments &moments) {
    return (moments.mean - expected_mean) /
           (std::sqrt(moments.variance) /
            std::sqrt(static_cast<double>(moments.n)));
}

double MaxErrorTolerance(double acceptance_probability) {
    double one_sided_pvalue = 0.5 * (1.0 - acceptance_probability);
    const double max_err = InverseNormalSurvival(one_sided_pvalue);
    ABEL_ASSERT(max_err > 0);
    return max_err;
}

}  // namespace random_internal

}  // namespace abel
